Options with extendable features have many applications in finance and these provide the motivation for this study. The pricing of extendable options when the underlying asset follows a geometric Brownian motion with constant volatility has appeared in the literature. In this paper, we consider holder-extendable call options when the underlying asset follows a mean-reverting stochastic volatility. The option price is expressed in integral forms which have known closed-form characteristic functions. We price these options using a fast Fourier transform, a finite difference method and Monte Carlo simulation, and we determine the efficiency and accuracy of the Fourier method in pricing holder-extendable call options for Heston parameters calibrated from the subprime crisis. We show that the fast Fourier transform reduces the computational time required to produce a range of holder-extendable call option prices by at least an order of magnitude. Numerical results also demonstrate that when the Heston correlation is negative, the Black–Scholes model under-prices in-the-money and over-prices out-of-the-money holder-extendable call options compared with the Heston model, which is analogous to the behaviour for vanilla calls.

中文翻译：

---

具有均值回归随机波动率的持有人可扩展看涨期权的定价

具有可扩展特性的选项在金融领域有很多应用，这些为这项研究提供了动力。当标的资产遵循具有恒定波动率的几何布朗运动时，可扩展期权的定价已经出现在文献中。在本文中，当标的资产遵循均值回归随机波动率时，我们考虑持有人可扩展看涨期权。期权价格以具有已知封闭形式特征函数的积分形式表示。我们使用快速傅里叶变换、有限差分法和蒙特卡洛模拟对这些期权进行定价，并确定傅里叶方法在为根据次贷危机校准的赫斯顿参数定价持有人可扩展看涨期权的效率和准确性。我们表明，快速傅里叶变换将产生一系列持有人可扩展看涨期权价格所需的计算时间减少了至少一个数量级。数值结果还表明，当 Heston 相关性为负时，与 Heston 模型相比，Black-Scholes 模型低估了价内持有人可延展看涨期权的价格，而高估了价外持有人可延展看涨期权，这是类似的。香草调用的行为。